where \(\kappa (\mu )\) , \(y_\nu (\mu )\) , and \(M_R(\mu )\)  depend on the energy scale \(\mu \)  being calculated up to a cutoff scale \(\Lambda \)We do not consider what occurs for scales above \(\Lambda \)  to generate mass for the right-handed neutrinos..
We have adopted the superscript notation to denote the different renormalization energy regions from the work of Antusch [1]}, for which the lowest region is denoted as \(a=1\) .
